Transcript Ch. 8 Fundamental of Hypothesis Testing

Ch. 9 Fundamental of Hypothesis
Testing
• There two types of statistical inferences, Estimation and
Hypothesis Testing
• Hypothesis Testing: A hypothesis is a claim (assumption)
about one or more population parameters.
– Average price of a six-pack in the U.S. is μ = $4.90
– The population mean monthly cell phone bill of this
city is: μ = $42
– The average number of TV sets in U.S. Homes is equal
to three; μ = 3
•
It Is always about a population parameter, not about a
sample statistic
•
Sample evidence is used to assess the probability that the
claim about the population parameter is true
A.
It starts with Null Hypothesis, H0
H 0 :μ  3
1.
2.
3.
and X=2.79
We begin with the assumption that H0 is true and any
difference between the sample statistic and true
population parameter is due to chance and not a real
(systematic) difference.
Similar to the notion of “innocent until proven guilty”
That is, “innocence” is a null hypothesis.
Null Hypo, Continued
4.
Refers to the status quo
5.
Always contains “=” , “≤” or “” sign
6.
May or may not be rejected
B.
1.
2.
3.
4.
5.
6.
Next we state the Alternative Hypothesis, H1
Is the opposite of the null hypothesis
1. e.g., The average number of TV sets in U.S. homes is
not equal to 3 ( H1: μ ≠ 3 )
Challenges the status quo
Never contains the “=” , “≤” or “” sign
May or may not be proven
Is generally the hypothesis that the researcher is trying to
prove. Evidence is always examined with respect to H1,
never with respect to H0.
We never “accept” H0, we either “reject” or “not reject”
it
Summary:
• In the process of hypothesis testing, the null hypothesis
initially is assumed to be true
• Data are gathered and examined to determine whether the
evidence is strong enough with respect to the alternative
hypothesis to reject the assumption.
• In another words, the burden is placed on the researcher to
show, using sample information, that the null hypothesis is
false.
• If the sample information is sufficient enough in favor of
the alternative hypothesis, then the null hypothesis is
rejected. This is the same as saying if the persecutor has
enough evidence of guilt, the “innocence is rejected.
• Of course, erroneous conclusions are possible, type I and
type II errors.
Reason for Rejecting H0
• Illustration: Let say, we assume that average age in the US
is 50 years (H0=50). If in fact this is the true (unknown)
population mean, it is unlikely that we get a sample mean
of 20. So, if we have a sample that produces an average of
20, them we reject that the null hypothesis that average
age is 50. (note that we are rejecting our assumption or
claim). (would we get 20 if the true population mean was
50? NO. That is why we reject 50)
How Is the Test done?
• We use the distribution of a Test Statistic, such as Z or t as
the criteria.
A. Rejection Region Method:
•
•
Divide the distribution into rejection and non-rejection
regions
Defines the unlikely values of the sample statistic if the
null hypothesis is true, the critical value(s)
–
•
Defines rejection region of the sampling distribution
Rejection region(s) is designated by  , (level of
significance)
–
Typical values are .01, .05, or .10
•
 is selected by the researcher at the beginning
•
 provides the critical value(s) of the test
Rejection Region or Critical Value Approach:
Level of significance = 
Non-rejection region
H0: μ = 12
H1: μ ≠ 12
a /2
a /2
Two-tail test
Represents
critical value
0
H0: μ ≤ 12
H1: μ > 12
a
0
Upper-tail test
H0: μ ≥ 12
H1: μ < 12
a
Lower-tail test
0
Rejection
region is
shaded
• P-Value Approach –
• P-value=Max. Probability of (Type I Error), calculated from
the sample.
• Given the sample information what is the size of blue are?
H0: μ = 12
H1: μ ≠ 12
Two-tail test
0
Upper-tail test
0
H0: μ ≤ 12
H1: μ > 12
H0: μ ≥ 12
H1: μ < 12
0
Type I and II Errors:
• The size of  , the rejection region, affects the risk of making different
types of incorrect decisions.
Type I Error
– Rejecting a true null hypothesis when it should NOT be rejected
– Considered a serious type of error
– The probability of Type I Error is 
– It is also called level of significance of the test
Type II Error
– Fail to reject a false null hypothesis that should have been rejected
– The probability of Type II Error is β
Decision
Actual Situation
Hypothesis
Legal System
Testing
H0 True
Do Not
Reject H0
Reject H0
No
Error
( 1  )
Type I
Error
( )
H0 False Innocence
Type II
Error
( )
No
Error
(1  )
No Error
(not guilty, found
not guilty)
( 1  )
Type I Error
(Not guilty,
found guilty)
( )
Not innocence
Type II Error
(guilty, found not
guilty)
( )
No Error
(guilty, found
guilty)
(1  )

Type I and Type II errors cannot happen at the same time
1.
Type I error can only occur if H0 is true
2.
Type II error can only occur if H0 is false
3.
There is a tradeoff between type I and II errors. If the
probability of type I error (  ) increased, then the
probability of type II error ( β ) declines.
4.
When the difference between the hypothesized
parameter and the actual true value is small, the
probability of type two error (the non-rejection
region) is larger.
5.
Increasing the sample size, n, for a given level of ,
reduces β
•
B.
P-Value approach to Hypothesis
Testing:
1.
The rejection region approach allows you to examine
evidence but restrict you to not more than a certain
probability (say  = 5%) of rejecting a true H0 by
mistake.
The P-value approach allows you to use the information
from the sample and then calculate the maximum
probability of rejecting a true H0 by mistake.
Another way of looking at P-value is the probability of
observing a sample information of “A=11.5” when the
true population parameter is “12=B”. The P-value is the
maximum probability of such mistake taking place.
2.
3.
4. That is to say that P-value is the smallest value
of  for which H0 can be rejected based on the
sample information
5. Convert Sample Statistic (e.g., sample mean) to
Test Statistic (e.g., Z statistic )
6. Obtain the p-value from a table or computer
7. Compare the p-value with 
–
If p-value <  , reject H0
–
If p-value   , do not reject H0
Test of Hypothesis for the Mean
σ known
The test statistic is:
X μ
Z 
σ
n
σ Unknown
The test statistic is:
t n-1
X μ

S
n
Steps to Hypothesis Testing
1. State the H0 and H1 clearly
2. Identify the test statistic (two-tail, one-tail, and Z
or t distribution
3. Depending on the type of risk you are willing to
take, specify the level of significance,
4. Find the decision rule, critical values, and
rejection regions. If –CV<actual value (sample
statistic) <+CV, then do not reject the H0
5.
Collect the data and do the calculation for the
actual values of the test statistic from the sample
Steps to Hypothesis testing, continued
Make statistical decision
Do not Reject H0
Conclude H0 may be
true
Make
management/business/
administrative decision
Reject H0
Conclude H1 is “true”
(There is sufficient evidence of H1)
When do we use a two-tail test?
when do we use a one-tail test?
• The answer depends on the question you are trying to
answer.
• A two-tail is used when the researcher has no idea which
direction the study will go, interested in both direction.
•
(example: testing a new technique, a new product, a new theory and we don’t know the
direction)
A new machine is producing 12 fluid once can of soft drink. The quality control
manager is concern with cans containing too much or too little. Then, the test is a twotailed test. That is the two rejection regions in tails is most likely (higher probability) to
provide evidence of H1.
H :   12oz
H :   12 oz
0
1
12
• One-tail test is used when the researcher is interested in the
direction.
• Example: The soft-drink company puts a label on cans
claiming they contain 12 oz. A consumer advocate desires
to test this statement. She would assume that each can
contains at least 12 oz and tries to find evidence to the
contrary. That is, she examines the evidence for less than
12 0z.
• What tail of the distribution is the most logical (higher
probability) to find that evidence? The only way to reject
the claim is to get evidence of less than 12 oz, left tail.
H :   12oz
H :   12 oz
0
1
11.5
12
14
Review of Hypo. Testing
• What is HT?
• Probability of making erroneous conclusions
• Type I – only when Null Hypo is true
• Type II – only when Null Hypo is false
• Two Approaches
• The Rejection or Critical Value Approach
• The P-value Approach (we calculate the observed level of significance)
• Test Statistics
• Z- distribution if Population Std. Dev. is Know
• t-distribution if the Population Std. Dev. is unknown
Rejection Region or Critical Value Approach:
The given level of significance = 
Non-rejection region
H0: μ = 12
H1: μ ≠ 12
a /2
a /2
Two-tail test
Represents
critical value
0
H0: μ ≤ 12
H1: μ > 12
a
0
Upper-tail test
H0: μ ≥ 12
H1: μ < 12
a
Lower-tail test
0
Rejection
region is
shaded
• P-Value Approach –
• P-value=Max. Probability of (Type I Error), calculated from
the sample.
• Given the sample information what is the size of the blue areas? (The
observed level of significance)
H0: μ = 12
H1: μ ≠ 12
Two-tail test
0
Upper-tail test
0
H0: μ ≤ 12
H1: μ > 12
H0: μ ≥ 12
H1: μ < 12
0
• Example 1:
• Let’s assume a sample of 25 cans produced a sample
mean of 11.5 0z and the population std dev=1 0z.
• Question 1:
– At a 5% level of significance (that is allowing for a
maximum of 5% prob. of rejecting a true null hypo), is
there evidence that the population mean is different from
12 oz?
• Null Hypo is:?
• Alternative Hypo is?
Can both approaches be used to answer this question?
A: Rejection region approach: calculate the actual test
statistics and compare it with the critical values
B: P-value approach: calculate the actual probability of
type I error given the sample information. Then
compare it with 1%, 5%, or 10% level of significance.
– Interpretation of Critical Value/Rejection Region
Approach:
– Interpretation of P-value Approach:
• Question 2:
– At a 5% level of significance (that is allowing for a
maximum of 5% prob. of rejecting a true null hypo), is
the evidence that the population mean is less than 12
oz?
• Null Hypo is:?
• Alternative Hypo is?
Can both approaches be used to answer this
question?
– Interpretation of Critical Value Approach:
– Interpretation of P-value Approach:
• Question 3:
– If in fact the pop. mean is 12 oz, what is the probability of obtaining a
sample mean of 11.5 or less oz (sample size 25)? Null
• Null Hypo is:?
• Alternative Hypo is?
• Question 4:
– If in fact the pop. mean is 12 oz, and the sample mean is 11.5 (or less),
what is the probability of erroneously rejecting the null hypo that the
pop. mean is 12 oz?
• Null Hypo is:?
• Alternative Hypo is?
Can both approaches be used to answer these question?
Connection to Confidence Intervals
• While the confidence interval estimation and hypothesis testing serve
different purposes, they are based on same concept and conclusions
reached by two methods are consistent for a two-tail test.
• In CI method we estimate an interval for the population mean with a
degree of confidence. If the estimated interval contains the
hypothesized value under the hypothesis testing, then this is equivalent
of not rejecting the null hypothesis. For example: for the beer sample
with mean 5.20, the confidence interval is:
P(4.61 ≤ μ ≤ 5.78)=95%
• Since this interval contains the Hypothesized mean ($4.90), we do not
(did not) reject the null hypothesis at  = .05
• Did not reject and within the interval, thus consistent results.